$$ \frac { \hat { 2 x - a y + a x } - 2 y } { x ^ { 2 } + ( 2 - y ) x - 2 y } = $$

Question

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Let's simplify the given rational expression:

$$
\frac{ \hat{2x - a y + a x} - 2 y }{ x^{2} + (2 - y) x - 2 y } = \, ?
$$

**Assumption:** It seems there might be a typographical error with the `\hat{}` notation. We'll proceed by interpreting the expression without the hat:

$$
\frac{2x - a y + a x - 2 y}{x^{2} + (2 - y) x - 2 y}
$$

### Step 1: Simplify the Numerator

Combine like terms in the numerator:

$$
2x - a y + a x - 2 y = (2x + a x) + (-a y - 2 y) = x(2 + a) - y(a + 2)
$$

Factor out $$(2 + a)$$:

$$
x(2 + a) - y(a + 2) = (2 + a)(x - y)
$$

So, the numerator simplifies to:

$$
(2 + a)(x - y)
$$

### Step 2: Factor the Denominator

The denominator is:

$$
x^{2} + (2 - y) x - 2 y
$$

Let's factor it:

1. **Look for two numbers that multiply to** $$1 \times (-2y) = -2y$$ **and add to** $$2 - y$$.

2. The numbers $$2$$ and $$-y$$ satisfy this condition:

- $$2 \times (-y) = -2y$$
   - $$2 + (-y) = 2 - y$$

3. Rewrite and factor by grouping:

$$
x^{2} + 2x - yx - 2y = x(x + 2) - y(x + 2) = (x - y)(x + 2)
$$

So, the denominator factors to:

$$
(x - y)(x + 2)
$$

### Step 3: Simplify the Fraction

Now, substitute the simplified numerator and denominator back into the fraction:

$$
\frac{(2 + a)(x - y)}{(x - y)(x + 2)}
$$

Assuming $$x \neq y$$ (to avoid division by zero), we can cancel out $$(x - y)$$:

$$
\frac{2 + a}{x + 2}
$$

### Final Simplified Expression

$$
\frac{2 + a}{x + 2}
$$

So, the original expression simplifies to:

$$
\frac{2 + a}{x + 2}
$$
The simplified expression is $$ \frac{2 + a}{x + 2} $$.

\( \frac { \hat { 2 x - a y + a x } - 2 y } { x ^ { 2 } + ( 2 - y ) x - 2 y } = \)

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